Understanding the application of Sobel kernel followed by ReLU to a zero-padded image
I assume that you would like to use convolution with padding with the same size of output matrix as this picture. You messed up calculations with convolution with "full padding". If we imagine these two matrices as windows that slide on each other, you can see that the filter is symmetrically inverted. I used a little bit different filter to show you in a better way how it works (I changed the last row to [-3, 0 3]).

Assuming these matrices:



You should add to your picture matrix two rows and columns of zeros:



Then you can start matrix multiplication, but notice that the filter is symmetrically inverted. The result shown as matrix $4 \times 4$ is the convolution with "full padding". The $2 \times 2$ matrix in the middle is a result of convolution with the "same padding", that you requested.



Next step:



Some iterations later:



And later:



And finally:



For convolution with the same size, the result will be the small $2 \times 2$ matrix in the middle.



After using the ReLU function, the result will be exactly the same.

So with using of your filter, the result would look like [[0,90], [0,90]].